WASHINGTON — Lawmakers sounded more optimistic on Friday than at any point since the 11-day shutdown began that they would soon end it, and that a more dangerous threat of a government default could be averted before Thursday’s deadline.

But they have so far failed to reach consensus on how to end the impasse that has crippled the government and sent Republican poll numbers plummeting — despite warmer feelings, White House meetings, congressional negotiations, and a slew of plans.

President Obama continued to increase his level of engagement in the process Friday, hosting Senate Republicans for 90 minutes at the White House and speaking by phone in the afternoon with House Speaker John A. Boehner.

Boehner’s spokesman and White House press secretary Jay Carney both said the dialogue would continue but did not provide further details of their call.

“The two of them agreed that all sides need to keep talking,” Carney said.

Although there are still sticking points, Carney called the discussions “constructive.”

“We’re obviously in a better place than we were a few days ago,” Carney said. “Everyone recognizes that default is not an option.”

Several of the 46 Republican senators who met with Obama at the White House on Friday described the meeting as free-flowing and valuable, but with no agreements.

“I don’t know how it’s going to play as far as the back and forth but I do think it’s going to play out to a successful conclusion fairly soon, over the next week,” said Senator Bob Corker, a Tennessee Republican who had been far more downcast in the days leading up to the shutdown.

House Republicans were offering a proposal to extend the nation’s borrowing authority for six weeks and quickly negotiate an end to the shutdown in exchange for agreements on spending and entitlement cuts that might replace the across-the-board cuts imposed earlier this year known as the sequester. Carney said the White House would sign legislation to temporarily raise the debt limit, but he also said the president was concerned about proposals to raise it only for six weeks — a time frame that would create economic uncertainty on the cusp of Thanksgiving and the start of holiday shopping season.

Senate Republicans, led by Susan Collins of Maine, were offering plans to end the twin crises in exchange for other agreements, including a repeal of a medical device tax that was levied to help pay for Obama’s health plan.

“It was constructive, but I don’t want to give the impression that he endorsed it,” Collins said after the meeting with Obama. “He said that there were elements of it that he thought work well.”

Lawmakers were preparing to meet during the holiday weekend. The Dow Jones industrial average rose by more than 100 points Friday following a 323-point jump Thursday as optimism about a debt ceiling deal spread.

House Republicans were scheduled to huddle Saturday morning. The Senate was also planning to vote to advance a Democratic bill to extend the debt ceiling until the end of 2014. Democrats, however, were not certain they had the 60 votes required to advance the bill.

“It looks to me like the two sides have inched a little closer to one another in the last 24 hours,” said Representative Tom Cole, an Oklahoma Republican who is close with House Speaker John Boehner. “I’d like them a step closer instead of an inch closer, but at least it’s moving in the right direction.”

Cole acknowledged the beating Republicans have taken in the polls, but said they were also being pressured by constituents because hundreds of thousands of government employees are no longer providing crucial services.

“Everybody is hearing some of that,” he said. “Hopefully that wakes everybody up around here.”

Treasury Secretary Jacob J. Lew has set a Thursday deadline to raise the $16.7 trillion debt limit to avoid a potential default, which could have worldwide economic consequences.

House Republicans, who met with Obama on Thursday night, said Friday that they were continuing to speak with the White House on a plan to offer a short-term extension of the nation’s borrowing authority, followed by faster track talks to reopen the government. Obama and Democrats have said they would like a long-term extension on the debt ceiling, with no strings attached, as well as a pact to open the government, before they negotiate about larger budget issues.

Conservative House members, who have driven much of the debate, may also be shifting their views. Representative John Fleming, a Louisiana Republican, said he would consider voting to raise the debt ceiling and reopen the government, and expects other Tea Party lawmakers would as well. Fleming said he would want assurances that Democrats would negotiate over two or three changes in entitlement programs in exchange for such a vote.

“It would depend on how comfortable I am that the president and Harry Reid are really open to substantive negotiations,” Fleming said. “I’m not interested in any cosmetics here.”

Globe staff writer Matt Viser contributed to this report. Noah Bierman can be reached at noah.bierman@globe.com.